We review fundamentals of Galerkin and conforming Finite Element (FE) methods using the model diffusion-convection-reaction problem. We discuss the possibility of different variational formulations leading to different energy spaces and corresponding conforming elements. The course is focusing on the famous inf-sup stability condition and the concept of discrete stability. We review the classical results of Babu ́ska, Mikhlin and Brezzi, and finish the exposition with fundamentals of the Discontinuous Petrov Galerkin (DPG) method. The week-long course consists of three 1.5 hour lectures per day accompanied with a one hour afternoon Q/A discussion session.